stay integer The number that cannot be divided by 2 is called odd [1] 。 In daily life, people usually call positive and odd numbers singular , it follows even numbers Is relative [2] 。 Odd numbers can be divided into positive odd numbers and negative odd numbers. The mathematical expression of odd number is: Positive and odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33
Negative odd numbers: - 1, - 3, - 5, - 7, - 9, - 11, - 13, - 15, - 17, - 19, - 21, - 23. - 25, - 27, - 29, - 31, - 33
With regard to odd and even numbers, there are the following properties:
(1) Two consecutive integer There must be an odd number and an even number in; (2) Odd+odd=even; Even+odd=odd; Even+Even++ Even=Even;
(3) Odd odd=even; Even odd=odd; Odd - even=odd;
(4) If a and b are integers, a+b and a-b have the same parity, that is, a+b and a-b are both odd or even numbers;
(5) The product of n odd numbers is odd, and the product of n even numbers is even; If one of the formulas is even, the product is even;
(6) The odd ones are 1, 3, 5, 7, 9; The even ones are 0, 2, 4, 6, 8;
(7) Divide the square of an odd number by 2, 4, or 8, and you have 1;
(8) The square difference of any two odd numbers is a multiple of 2, 4, 8;
(9) The odd number is divided by 2 and the remainder is 1.
Famous mathematician Pythagoras Interesting odd number phenomenon is found: add odd numbers consecutively, and the result of each time is just the square number. This is reflected in odd numbers and Square number There are close and important links between them. For example: Property: Any odd number can be written as the square difference of two integers.
① For example, 1=1 ² - 0 ², 3=2 ² - 1 ², 5=3 ² - 2 ²
Let the positive odd number a be the nth positive odd number (that is, n ≥ 1), then a=n ² - (n-1) ²=2n-1; a=(a+1-n)²-(a-n)²=2a-2n+1。
② For example, - 1=0 ² - 1 ², - 3=1 ² - 2 ², - 5=2 ² - 3 ²
Let the negative odd number b be the nth negative odd number (n ≥ 1), change the sign by ①, and it is easy to get b=- a=(n-1) ² - n ²=1-2n;
But the second rule is different from positive odd numbers.
Odd and prime number Odd numbers may or may not be prime numbers. For example, 3 is an odd number and a prime number; 9 is odd, but not prime. Three prime number theorem: Each odd number Can be expressed as the sum of three prime numbers. [3] series ：1，3，5，7，9，…… ，2n-1，... It is called odd number column, and the general formula is 。 It has a beautiful property: when n takes any positive integer, the sum of its first n terms is a Perfect square , i.e 。 Odd number columns can also be expressed from another perspective: if ， , when Both , then sequence Is an odd number column. [4] 
